Let us clarify our objective. It suffices to prove that the \(L^p\)  Regularity problem for the original operator \(\mathcal {L}=\mathrm {div}(A\nabla \cdot )\)  is solvable for at least one value \(q\in (1,\infty )\)  as then by [1]} it follows that \(L^p\)  Regularity problem for \(\mathcal {L}\)  is solvable if and only if the \(L^{p^{\prime }}\)  Dirichlet problem is solvable for the adjoint operator \(\mathcal {L}^*\) . (See Theorem 1.1 of [1]}) But solvability of Dirichlet problem satisfying Carleson condition in the range \((p_{dir},\infty )\)  for some \(p_{dir}>1\)  has been resolved in [3]}, and hence the claim about the range of solvability of Regularity stated in Theorem REF  would follow.
